Trigonometric Functions And Equations Notes

Introduction

Trigonometric functions and equations form a fundamental part of JEE Main Mathematics. This topic is not only crucial for scoring high in the exam but also forms the basis for many advanced concepts in calculus, algebra, and geometry. In this study note, we will break down the complex ideas into smaller, manageable sections, explain each part clearly, and provide examples where necessary to make the concepts digestible.

Trigonometric Functions

Definition and Basic Functions

Trigonometric functions are functions of an angle that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are:

• Sine ($\sin$)
• Cosine ($\cos$)
• Tangent ($\tan$)
• Cosecant ($\csc$)
• Secant ($\sec$)
• Cotangent ($\cot$)

These functions are defined as follows for an angle $\theta$ in a right-angled triangle:

$$ \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} $$

$$ \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} $$

$$ \tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{\sin \theta}{\cos \theta} $$

$$ \csc \theta = \frac{1}{\sin \theta} $$

$$ \sec \theta = \frac{1}{\cos \theta} $$

$$ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} $$

Unit Circle Representation

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is a powerful tool for understanding trigonometric functions.

• The $x$-coordinate of a point on the unit circle is $\cos \theta$.
• The $y$-coordinate of a point on the unit circle is $\sin \theta$.

For any angle $\theta$, the coordinates $(\cos \theta, \sin \theta)$ represent the point on the unit circle corresponding to that angle.

Diagram: Unit Circle showing angles in radians and corresponding coordinates

Periodicity and Symmetry

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The periodicities are:

• $\sin \theta$ and $\cos \theta$ have a period of $2\pi$.
• $\tan \theta$ and $\cot \theta$ have a period of $\pi$.

Symmetry properties:

• $\sin(-\theta) = -\sin \theta$ (Odd function)
• $\cos(-\theta) = \cos \theta$ (Even function)
• $\tan(-\theta) = -\tan \theta$ (Odd function)

Trigonometric Identities

Fundamental Identities

1. Pythagorean Identities:

$$ \sin^2 \theta + \cos^2 \theta = 1 $$

$$ 1 + \tan^2 \theta = \sec^2 \theta $$

$$ 1 + \cot^2 \theta = \csc^2 \theta $$

1. Angle Sum and Difference Identities:

$$ \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b $$

$$ \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b $$

$$ \tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} $$

Double Angle and Half Angle Identities

1. Double Angle Identities:

$$ \sin 2\theta = 2 \sin \theta \cos \theta $$

$$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta $$

$$ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} $$

1. Half Angle Identities:

$$ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} $$

$$ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} $$

$$ \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta} $$

Trigonometric Equations

Solving Basic Trigonometric Equations

To solve trigonometric equations, we often use identities and algebraic manipulation. Consider the equation:

$$ \sin x = \frac{1}{2} $$

The general solutions are:

$$ x = n\pi + (-1)^n \frac{\pi}{6}, \quad n \in \mathbb{Z} $$

Using Inverse Trigonometric Functions

Inverse trigonometric functions help in finding angles when the values of trigonometric functions are known. For example:

$$ \sin^{-1}(x), \cos^{-1}(x), \tan^{-1}(x) $$

If $\sin \theta = x$, then $\theta = \sin^{-1}(x)$.

Conclusion

Trigonometric functions and equations are vital for mastering JEE Main Mathematics. Understanding the definitions, unit circle, identities, and methods to solve equations is crucial. Practice regularly with a variety of problems to gain confidence and proficiency.